Question: How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$?
Answer: Since $|m| > 0$, we can clear fractions from the inequalities, arriving at $8 \geq |m|$.  This is satisfied for $-8 \leq m \leq 8$.  There are 17 integers in this range, but 0 is not allowed, so our final answer is $\boxed{16}$.